Question about Cartesian Tensors

[Worn_Out_Tools]

I am not a mathematician but an Engineer-in-Training studying mechanics. That being said why does equation I-10 equal 1 when k equals i and 0 when k does not equal i?

 

[Hill]

I am not a mathematician but an Engineer-in-Training studying mechanics. That being said why does equation I-10 equal 1 when k equals i and 0 when k does not equal i?

Consider the case of $x'_0=1, x'_1=0, x'_2=0$. In this case, only the terms with $k=0$ contribute to the sum in the eqn. I-9, which becomes $x'_i=a_{ij}a_{0j}$. For $i=0$ it becomes $1=a_{0j}a_{0j}$, for $i=1$ it becomes $0=a_{1j}a_{0j}$, etc. For all combinations of $k$ and $i$, you get the eqn. I-10.

 

[Worn_Out_Tools]

“a” is defined as shorthand for the cosine function earlier in the book. So 1 = cos(0,j) * cos (0,j)?

 

[Hill]

“a” is defined as shorthand for the cosine function earlier in the book. So 1 = cos(0,j) * cos (0,j)?

No. $i, j, k$ are indices.